Blasting two databases

I assume you mean two sets of sequences since databases can't be blasted against each other.

This can be easily done using blat (LINK Jim Kent from UCSC) or blast+ if you prefer that. With blat you would not need to create database indexes, just search with one file against the other. With blast+ you will need to create indexes for one and use that as the database to search against.

Use a tabular output format like 6 or 7 to make parsing easier later.

Note: blat would be useful only if your sequences are reasonably similar. If they are not then you should use blast+.

You can do a reciprocal blast search (which I think is intended here) using a tool like proteinortho https://www.bioinf.uni-leipzig.de/Software/proteinortho/